The simple absorption model describes the motion of a defect on a periodically replicated square lattice of size \(\mathrm {L}\)  and coordination number \(Z=4\) . The defect hops from any site to any of its four nearest neighboring sites with reduced frequency of 1. The number of transient states is \(N=\mathrm {L}^2\) . In addition, the defect can reach the absorbing sink from a singularized site with an absorbing frequency equal to \(10^{-2}\) . In this particular application, the considered transition rate matrix \(\mathbf {A}\)  is thus a modified Laplacian matrix: diagonal elements are equal to 4, but one element is set to value \(4.01\) , while the \(4N\)  off-diagonal elements corresponding to transitions are all equal to -1. The matrix \(\mathbf {A}\)  is thus symmetric positive definite by construction. We herein evaluate the cost of extracting the linear system using the various sparse iterative solvers from SLEPC library [1]} in PETSC environment [2]} and with varying the number of cores in the computations. This enables one to deduce the speedup and efficiency resulting from implementing the solvers on a parallel computer architecture.
